Add new commit to the existing Git tag

You can’t put a new commit into an existing tag without breaking an important Git guideline: Never(*) modify commits that you have published.

Tags in Git aren’t meant to be mutable. Once you push a tag out there, leave it alone.

You can, however, add some changes on top of v1.1 and release something like v1.1.1 or v1.2. One way of doing that would be

# Create a new branch from tag v1.1
git checkout -b newbranch v1.1

# Do some work and commit it

# Create a new tag from your work
git tag -a -m "Tag version 1.1.1, a bugfix release" v1.1.1

(*) Unless you have a really super special reason for doing so, and only if you completely understand the implications, and even then, don’t make a habit of it.